A park planner is designing a triangular walking trail connecting three landmarks: a fount… — ACT

A park planner is designing a triangular walking trail connecting three landmarks: a fountain at point $F$ , a gazebo at point $G$ , and a bridge at point $B$ . The distance from the fountain to the gazebo is $800$ meters, and the distance from the gazebo to the bridge is $1{}500$ meters. If the angle formed at the gazebo ( $\angle FGB$ ) measures $60^\circ$ , what is the direct distance, in meters, from the fountain at point $F$ to the bridge at point $B$ ?

Cevap: 1300 meters

Cevap

1300 meters

Applying the Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos(C)

with

a = 800

b = 1500

, and

C = 60^\circ

gives

c^2 = 800^2 + 1500^2 - 2(800)(1500)(0.5) = 1{}690{}000

. Taking the square root yields

c = 1300

meters.

Adım Adım Çözüm

Identify given measurements and formula

FG = 800

GB = 1500

\angle FGB = 60^\circ

, using Law of Cosines

c^2 = a^2 + b^2 - 2ab\cos(C)

Two side lengths and the included angle (SAS) are known, requiring the Law of Cosines to solve for the opposite side.

Substitute values into the Law of Cosines equation

FB^2 = 800^2 + 1500^2 - 2(800)(1500)\cos(60^\circ)

Direct replacement of known side lengths and angle measure into the formula.

Evaluate the arithmetic terms

FB^2 = 640{}000 + 2{}250{}000 - 1{}200{}000 = 1{}690{}000

Squaring the side lengths and calculating the product

2(800)(1500)(0.5)

Take the square root to solve for

FB

FB = \sqrt{1{}690{}000} = 1300

Taking the principal square root yields the distance in meters.

Anahtar Kavram

Law of Cosines (SAS Triangle Solving)

Soru

Cevap

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Anahtar Kavram

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